Average Error: 6.9 → 0.7
Time: 1.5m
Precision: 64
\[\frac{x \cdot 2.0}{y \cdot z - t \cdot z}\]
↓
\[\begin{array}{l} \mathbf{if}\;y \cdot z - t \cdot z \le -1.229546345171507 \cdot 10^{+160}:\\ \;\;\;\;\frac{2.0 \cdot \frac{x}{z}}{y - t}\\ \mathbf{elif}\;y \cdot z - t \cdot z \le -2.7459824140421896 \cdot 10^{-246}:\\ \;\;\;\;\frac{x \cdot 2.0}{y \cdot z - t \cdot z}\\ \mathbf{elif}\;y \cdot z - t \cdot z \le 1.5369969526822363 \cdot 10^{-164}:\\ \;\;\;\;\frac{2.0 \cdot \frac{x}{z}}{y - t}\\ \mathbf{elif}\;y \cdot z - t \cdot z \le 5.071348037924594 \cdot 10^{+164}:\\ \;\;\;\;\frac{x \cdot 2.0}{y \cdot z - t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.0 \cdot \frac{x}{z}}{y - t}\\ \end{array}\]
\frac{x \cdot 2.0}{y \cdot z - t \cdot z}
↓
\begin{array}{l}
\mathbf{if}\;y \cdot z - t \cdot z \le -1.229546345171507 \cdot 10^{+160}:\\
\;\;\;\;\frac{2.0 \cdot \frac{x}{z}}{y - t}\\

\mathbf{elif}\;y \cdot z - t \cdot z \le -2.7459824140421896 \cdot 10^{-246}:\\
\;\;\;\;\frac{x \cdot 2.0}{y \cdot z - t \cdot z}\\

\mathbf{elif}\;y \cdot z - t \cdot z \le 1.5369969526822363 \cdot 10^{-164}:\\
\;\;\;\;\frac{2.0 \cdot \frac{x}{z}}{y - t}\\

\mathbf{elif}\;y \cdot z - t \cdot z \le 5.071348037924594 \cdot 10^{+164}:\\
\;\;\;\;\frac{x \cdot 2.0}{y \cdot z - t \cdot z}\\

\mathbf{else}:\\
\;\;\;\;\frac{2.0 \cdot \frac{x}{z}}{y - t}\\

\end{array}
double f(double x, double y, double z, double t) {
        double r23358729 = x;
        double r23358730 = 2.0;
        double r23358731 = r23358729 * r23358730;
        double r23358732 = y;
        double r23358733 = z;
        double r23358734 = r23358732 * r23358733;
        double r23358735 = t;
        double r23358736 = r23358735 * r23358733;
        double r23358737 = r23358734 - r23358736;
        double r23358738 = r23358731 / r23358737;
        return r23358738;
}
↓
double f(double x, double y, double z, double t) {
        double r23358739 = y;
        double r23358740 = z;
        double r23358741 = r23358739 * r23358740;
        double r23358742 = t;
        double r23358743 = r23358742 * r23358740;
        double r23358744 = r23358741 - r23358743;
        double r23358745 = -1.229546345171507e+160;
        bool r23358746 = r23358744 <= r23358745;
        double r23358747 = 2.0;
        double r23358748 = x;
        double r23358749 = r23358748 / r23358740;
        double r23358750 = r23358747 * r23358749;
        double r23358751 = r23358739 - r23358742;
        double r23358752 = r23358750 / r23358751;
        double r23358753 = -2.7459824140421896e-246;
        bool r23358754 = r23358744 <= r23358753;
        double r23358755 = r23358748 * r23358747;
        double r23358756 = r23358755 / r23358744;
        double r23358757 = 1.5369969526822363e-164;
        bool r23358758 = r23358744 <= r23358757;
        double r23358759 = 5.071348037924594e+164;
        bool r23358760 = r23358744 <= r23358759;
        double r23358761 = r23358760 ? r23358756 : r23358752;
        double r23358762 = r23358758 ? r23358752 : r23358761;
        double r23358763 = r23358754 ? r23358756 : r23358762;
        double r23358764 = r23358746 ? r23358752 : r23358763;
        return r23358764;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.9
Target2.0
Herbie0.7
\[\begin{array}{l} \mathbf{if}\;\frac{x \cdot 2.0}{y \cdot z - t \cdot z} \lt -2.559141628295061 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2.0\\ \mathbf{elif}\;\frac{x \cdot 2.0}{y \cdot z - t \cdot z} \lt 1.045027827330126 \cdot 10^{-269}:\\ \;\;\;\;\frac{\frac{x}{z} \cdot 2.0}{y - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - t\right) \cdot z} \cdot 2.0\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (- (* y z) (* t z)) < -1.229546345171507e+160 or -2.7459824140421896e-246 < (- (* y z) (* t z)) < 1.5369969526822363e-164 or 5.071348037924594e+164 < (- (* y z) (* t z))

    1. Initial program 13.5

      \[\frac{x \cdot 2.0}{y \cdot z - t \cdot z}\]

    2. Simplified1.2

      \[\leadsto \color{blue}{\frac{\frac{2.0}{\frac{z}{x}}}{y - t}}\]

    3. Taylor expanded around 0 1.1

      \[\leadsto \frac{\color{blue}{2.0 \cdot \frac{x}{z}}}{y - t}\]

    if -1.229546345171507e+160 < (- (* y z) (* t z)) < -2.7459824140421896e-246 or 1.5369969526822363e-164 < (- (* y z) (* t z)) < 5.071348037924594e+164

    1. Initial program 0.3

      \[\frac{x \cdot 2.0}{y \cdot z - t \cdot z}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z - t \cdot z \le -1.229546345171507 \cdot 10^{+160}:\\ \;\;\;\;\frac{2.0 \cdot \frac{x}{z}}{y - t}\\ \mathbf{elif}\;y \cdot z - t \cdot z \le -2.7459824140421896 \cdot 10^{-246}:\\ \;\;\;\;\frac{x \cdot 2.0}{y \cdot z - t \cdot z}\\ \mathbf{elif}\;y \cdot z - t \cdot z \le 1.5369969526822363 \cdot 10^{-164}:\\ \;\;\;\;\frac{2.0 \cdot \frac{x}{z}}{y - t}\\ \mathbf{elif}\;y \cdot z - t \cdot z \le 5.071348037924594 \cdot 10^{+164}:\\ \;\;\;\;\frac{x \cdot 2.0}{y \cdot z - t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{2.0 \cdot \frac{x}{z}}{y - t}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (x y z t)
  :name "Linear.Projection:infinitePerspective from linear-1.19.1.3, A"

  :herbie-target
  (if (< (/ (* x 2.0) (- (* y z) (* t z))) -2.559141628295061e-13) (* (/ x (* (- y t) z)) 2.0) (if (< (/ (* x 2.0) (- (* y z) (* t z))) 1.045027827330126e-269) (/ (* (/ x z) 2.0) (- y t)) (* (/ x (* (- y t) z)) 2.0)))

  (/ (* x 2.0) (- (* y z) (* t z))))